Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-02T21:01:17.839Z Has data issue: false hasContentIssue false
  • English
  • Français

Associativity of the regular semidirect product of existence varieties

Part of: Semigroups

Published online by Cambridge University Press:  09 April 2009

Bernd Billhardt  and
Mária B. Szendrei
Bernd Billhardt
Affiliation:
Universität-Gesamthochschule KasselFachbereich 17 Mathematik/Informatik D–34109 Kassel Holländische Str. 36Germany e-mail: [email protected]
Mária B. Szendrei
Affiliation:
University of Szeged Bolyai InstituteH–6720 Szeged Aradi vértanúk tere 1Hungary e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The associativity of the regular semidirect product of existence varieties introduced by Jones and Trotter was proved under certain condition by Reilly and Zhang. Here we estabilsh associativity in many new cases. Moreover, we prove that the regular semidirect product is right distributive with respect to the join operation. In particular, both associativity and right distriutivity yiel within the varieties of completely simple semigroups. Analogous results are obtainedj for e-pseudovarieties of finite regular semigroups.

MSC classification

Secondary: 20M17: Regular semigroups 20M07: Varieties and pseudovarieties of semigroups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 69 , Issue 1 , August 2000 , pp. 85 - 115
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Auinger, K. and Trotter, P. G., ‘Pseudovarieties, regular semigroups and semidirect products’, J. London Math. Soc. (2) 58 (1998), 284–296.CrossRefGoogle Scholar
[2]Billhardt, B., ‘On λ-semidirect products by locally R-unipotent semigroups’, Acta Sci. Math., to appear.Google Scholar
[3]Billhardt, B., ‘On a wreath product embedding and idempotent pure congruences on inverse semigroups’, semigroup Forum 45 (1992), 45–54.CrossRefGoogle Scholar
[4]Eilenberg, S., Automata, languages and machines, Vol. B (Academic Press, Berlin, 1976).Google Scholar
[5]Hall, T. E., ‘Identities for existence varieties of regular semigroups’, Bull. Austral. Math. Soc. 40 (1989), 59–77.CrossRefGoogle Scholar
[6]Houghton, C. H., ‘Embedding inverse semigroups in wreath products’, Glasgow Math. J. 17 (1976), 77–82.CrossRefGoogle Scholar
[7]Howie, J. M., FUndamentals of semigroups theory, London Math. Soc. Monographs, New Series 12 (Oxford University Press, Oxford, 1995).CrossRefGoogle Scholar
[8]Jones, P. R., ‘Rees Matrix covers and semidirct products of regular semigroups’, J. Algebra 218 (1999), 287–306.CrossRefGoogle Scholar
[9]Jones, P. R. and Trotter, P. G., ‘Semidirect products of regular semigroups’, Trans. Amer. Math. Soc. 349 (1997), 4265–4310.CrossRefGoogle Scholar
[10]Kad'ourek, J., ‘On some existence varieties of locally orthodox semigroups’, Internat. J. Algebra Comput. 7 (1997), 93–131.CrossRefGoogle Scholar
[11]Kad'ourek, J. and Szendrei, M. B., ‘A new approach in the theory of orthodox semigroups’, Semigroup Forum 40 (1990), 257–296.CrossRefGoogle Scholar
[12]Mangold, M., E-varieties and e-pseudovarieties of regular semigroups (Ph.D. Thesis, Monash University, 1995).Google Scholar
[13]Meldrum, J. D. P., Wreath products of groups and semigroups Pitman Monographs Surveys Pure appl. Math. 74 (Longman, Essex, 1995).Google Scholar
[14]Neumann, H., Varieties of groups (Springer, Berlin, 1967).CrossRefGoogle Scholar
[15]Reilly, N. R. and Zhang, S., ‘Associativity of products of existence varieties of regular semigroups’, J. Pure Appl. Algebra 135 (1999), 57–91.CrossRefGoogle Scholar
[16]Yeh, Y. T., ‘The existence of e-free objects in e-varieties of regular semigroups’, Internat. J. Algebra Comput. 2 (1992), 471–484.CrossRefGoogle Scholar